Math, asked by creedezio17, 8 months ago

Simplify this question

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Answers

Answered by RvChaudharY50
18

Qᴜᴇsᴛɪᴏɴ :-

Simplify :- 1/√[ 7 - 4√3]

Sᴏʟᴜᴛɪᴏɴ :-

→ 1/√[ 7 - 4√3]

Lets Try to Solve Denominator Root Part First ,

→ 7 - 4√3

Splitting The Terms, we can write ,

→ 4 + 3 - 2 * 2 * √3

→ (2)² + (√3)² - 2 * 2 * √3

comparing it with a² + b² - 2ab = (a - b)² we get,

→ (2 - √3)²

So,

√[2 - √3]²

we know that, (√a)² = a ,

→ (2 - √3)

Now, Our Question becomes :- 1/(2 - √3)

→ 1/(2 - √3)

Rationalizing The denominator we get,

→ 1/(2 - √3) * (2 + √3)/(2 + √3)

using (a + b)(a - b) = a² - b² in Denominator now,

→ (2 + √3) / [(2)² - (√3)²]

→ (2 + √3) / [ 4 - 3 ]

→ (2 + √3) / 1

→ (2 + √3) (Ans.)

Answered by Anonymous
15

{\huge{\bf{\red{\underline{Solution:}}}}}

{\bf{\blue{\underline{Given:}}}}

 \star \: {\tt{ \frac{1}{ \sqrt{7 - 4 \sqrt{3} } } }}\\

{\sf{\underline{\blue{Now}}}}

\implies{\tt{ \frac{1}{  \big(\sqrt{7 - 4 \sqrt{3} \big)}} }}\\ \\

\implies{\tt{   \frac{1}{ \sqrt{7 - 2 \times 2 \times  \sqrt{3} } }  }}\\ \\

\implies{\tt{   \frac{1}{ \sqrt{4 + 3 - 2 \times 2 \times  \sqrt{3} } }  }}\\ \\

\implies{\tt{   \frac{1}{ \sqrt{ {(2)}^{2} + ( \sqrt{3^{2} }  )- 2 \times 2 \times  \sqrt{3} } }  }}\\ \\

 \star \: \boxed{\tt{ \purple{ ({x}  -   {y})^{2}  = {x}^{2}   +  {y}^{2} - 2xy  }}}\\ \\

where x=2 and y=√3

\implies{\tt{    \frac{1}{  \sqrt{(2 -  \sqrt{3}  )^{2}} }  }}\\ \\

{\sf{\underline{\blue{Rationalizing}}}}\\ \\

 \implies{\tt{    \frac{1}{  (2 -  \sqrt{3}  )} \times  \frac{(2 +  \sqrt{3} )}{(2 +  \sqrt{3} )}   }}\\ \\

\implies{\tt{    \frac{(2 +  \sqrt{3}) }{  (2 -  \sqrt{3}  )(2 +  \sqrt{3} )}  }}\\ \\

\star \boxed{\tt{ \purple{ {x}^{2}  -  {y}^{2}  = (x - y)(x + y)}}}\\ \\

\implies{\tt{    \frac{(2 +  \sqrt{3}) }{   {(2)}^{2}  - ( \sqrt{3} )^{2} }  }}\\ \\

 \implies{\tt{     \frac{2 +  \sqrt{3} }{4 - 3}  }}\\ \\

 \implies \boxed{\tt{    2 +  \sqrt{3} \:Ans }}

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